Postulates & Theorems

Front 1

ruler postulate

Back 1

1. the points on a line can be paired with the real numbers in such a way that any points can have coordinates 0 and 1.

2. once a coordinate system has been chosen in this way, the distance between any two points equals the absolute value of the difference of their coordinates.

Front 2

segment addition postulate

Back 2

if B is between A and C, then AB + BC = AC.

Front 3

angle addition postulate

Back 3

if point B lies in the interior of <AOC, then m<AOB + m<BOC = m<AOC. If <AOC is a straight angle and B is any point not on line AC, then m<AOB + m<BOC = 180.

Front 4

what's in a line? futhermore- what's in a plane? what's in space?

Back 4

a line contains at least two points; a plane contains at least three points not all in one line; space contains at least four points not all in one plane.

Front 5

break on through, through the other line.

Back 5

through any two points there is exactly one line.

Front 6

plane points postulate

Back 6

through any three points there is at least one plane, and through any three noncollinear points there is exactly one plane.

Front 7

2 points walk into plane...

Back 7

if two points are in a plane, then the line that contains the points is in that plane.

Front 8

it's just plane intersection.

Back 8

if two planes intersect, then their intersection is a line.

Front 9

trans slash!

Back 9

if two parallel lines are cut by a transversal, then corresponding angles are congruent.

Front 10

it's very effective.

Back 10

if two lines are cut by a transversal and corresponding angles are congruent, then the lines are parallel.

Front 11

SSS Postulate

Back 11

if three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent.

Front 12

SAS Postulate

Back 12

if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent.

Front 13

ASA Postulate

Back 13

if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent.

Front 14

AA Similarity Postulate

Back 14

if two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.

Front 15

point of intersection theorem

Back 15

if two lines intersect, then they intersect in exactly one point.

Front 16

you're out of line, BUB!

Back 16

through a line and a point not in the line there is exactly one plane.

Front 17

line containment

Back 17

if two lines intersect, then exactly one plane contains the lines.

Front 18

midpoint theorem

Back 18

if M is the point of line segment AB, then AM = 1/2 AB and MB = 1/2 AB

Front 19

angle bisector theorem

Back 19

if ray BX is the bisector <ABC, then m<ABX = 1/2 m<ABC and m<XBC = 1/2 m<ABC.

Front 20

vertical angles

Back 20

vertical angles are congruent

Front 21

adjacent angles/ perp lines

Back 21

if two lines are perpendicular, then they form congruent adjacent angles.

Front 22

converse of perp lines/ adjacent angles

Back 22

if two lines form congruent adjacent angles, then the lines are perpendicular

Front 23

XXXterior adjaycent angles

Back 23

if the exterior sides of adjacent acute angles are perpendicular, then the angles are complementary.

Front 24

supplements of congruent angles

Back 24

if two angles are supplements of congruent angles (or of the same angle), then the two angles are congruent.

Front 25

complements of congruent angles

Back 25

if two angles are complements of congruent angles (or of the same angle), then the two angles are congruent.

Front 26

when parallel planes are cutcutcut

Back 26

if two parallel planes are cut by a third plane, then the lines of the intersection are parallel.

Front 27

alt int anglez

Back 27

if two parallel lines are cut by a transversal, then alternate interior angles are congruent.

Front 28

same side int.

Back 28

if two parallel lines are cut by a transversal, then same-side interior angles are supplementary

Front 29

perp. trans. w/ parallel lines

Back 29

in a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other one also

Front 30

converse of alt int angles

Back 30

if two lines are cut by a transversal and alternate interior angles are congruent, then the lines are parallel

Front 31

converse of same-side angles

Back 31

if two lines are cut by a transversal and same-side interior angles are supplementary, then the lines are parallel.

Front 32

in a plane.... two lines, same lines (perp!!)

Back 32

in a plane two lines perpendicular to the same line are parallel.

Front 33

thru a point outside a line... given line (2)

Back 33

through a point outside a line, there is exactly one line parallel to the given line.

through a point outside a line, there is exactly one line perpendicular to the given line.

Front 34

2 parallel lines, then 3?????? THEN 1000000

Back 34

two lines parallel to a third line are parallel to each other.

Front 35

triangle total angle sum

Back 35

the sum of the measures of the angles of a triangle is 180.

Front 36

exterior angle= these two bb anglez

Back 36

the measure of an exterior angle of a triangle equals the sum of the measures of the two remote interior angles.

Front 37

sum of angle measures of convex polygon

Back 37

the sum of the measures of the angles of a convex polygon with n sides is (n-2)180.

Front 38

sum of the measures of exterior angles of convex polygon

Back 38

the sum of the measures of the exterior angles of any convex polygon, one angle at each vertex, is 360.

Front 39

isosceles triangle theorem

Back 39

if two sides of a triangle are congruent, then the angles opposite those sides are congruent.

*corollary- the bisector of the vertex of an isosceles triangle is perpendicular to the base at its midpoint

Front 40

kinda like isos. triangle thm but not.

Back 40

if two angles of a triangle are congruent, then the sides opposite those angles are congruent.

Front 41

AAS Theorem

Back 41

if two angles and a non-included side of one triangle are congruent to the corresponding parts of another triangle, then the triangles are congruent.

Front 42

HL Theorem

Back 42

if the hypotenuse and a leg of one triangle are congruent to the corresponding parts of another right triangle, then the triangles are congruent.

Front 43

a point walks into a perp. bisector. then what?

Back 43

if a point lies on the perpendicular bisector of a segment, then the point is equidistant from the endpoints of the segment.

Front 44

equidistant point from segment endpoints

Back 44

if a point is equidistant from the endpoints of the segment, then the points lie on the perpendicular bisector of the segment.

Front 45

a point walks into a bisector of an angle. then what?

Back 45

if a point lies on the bisector of an angle, then the point is equidistant from the sides of the angle.

Front 46

equidistant point from angle sides

Back 46

if a point is equidistant from the sides of the angle, then the point lies on the bisector of an angle.

Front 47

properties of parallelograms

Back 47

-opp. sides congruent

-opp. angles congruent

-diagonals bisect each other

Front 48

opp sides of quad are congruent

Back 48

if both pair of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram

Front 49

opp sides congruent and parallel... more like parahell amiright lelelele

Back 49

if one pair of opposite sides of a quadrilateral are both congruent and parallel, then the quadrilateral is a parallelogram.

Front 50

both quad opp angles congruent

Back 50

if both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram.

Front 51

diagonals bisect each other in a bar i mean a quadrilateral

Back 51

if the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.

Front 52

parallel lines/ points on one line

Back 52

if two lines are parallel, then all the points on one line are equidistant from the other line.

Front 53

congruent segs cut off

Back 53

if three parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal.

Front 54

a line that contains a midpoint of one side of a triangle

Back 54

a line that contains the midpoint of one side of a triangle and is parallel to another side passes through the midpoint of the third side.

Front 55

seg that joins the midpoint

Back 55

the segment that joins the midpoints of two sides of a triangle

1) is parallel to the third side

2) is half as long as the third side

Front 56

rectangle!

Back 56

the diagonals of a rectangle are congruent

Front 57

rhombus

Back 57

-diagonals of a rhombus are perpendicular

-each diagonal bisects two angles of the rhombus

Front 58

hypotenuse midpoint!

Back 58

the midpoint of the hypotenuse of a right triangle is equidistant from the three vertices.

Front 59

if a parallelogram has right angle

Back 59

if an angle of a parallelogram is a right angle, then the parallelogram is a rectangle

Front 60

consecutive parallelogram sides

Back 60

if two consecutive sides of a parallelogram are congruent, then the parallelogram is a rhombus.

Front 61

base isos. trap. angles

Back 61

base angles of an isosceles trapezoid are congruent

Front 62

median of trap

Back 62

the median of a trapezoid

1) is parallel to the bases

2) has a length equal to the average of the base lengths

Front 63

exterior angle inequality theorem

Back 63

the measure of an exterior angle of a triangle is greater than the measure of either remote interior angle.

Front 64

side inequalities in a triangle, their relation to angles

Back 64

if one side of a triangle is longer than a second side, then the angle opposite the first side is larger than the angle opposite the second side.

Front 65

angle inequalities in a triangle, their relation to sides.

Back 65

if one angle of a triangle is larger than the second angle, then the side opposite the first angle is longer than the side opposite the second angle.

Front 66

the triangle inequality

Back 66

the sum of the lengths of any sides of a triangle is greater than the length of the third side

Front 67

SAS inequality theorem

Back 67

if two sides of one triangle are congruent to two sides of another triangle, but the included angle of the first triangle is larger than the included angle of the second, then the third side of the first triangle is longer than the third side of the second triangle.

Front 68

SSS inequality theorem

Back 68

if two sides of one triangle are congruent to two sides of another triangle, bu the third side of the first triangle is longer than the third side of the second, then the included angle of the first triangle is larger than the included angle of the second.

Front 69

SAS ~

Back 69

if an angle of one triangle is congruent to an angle of another triangle and the sides including those angles are in proportion, then the triangles are similar.

Front 70

SSS ~

Back 70

if the sides of two triangles are in proportion, then the triangles are similar.

Front 71

triangle proportionality theorem

Back 71

if a line parallel to one side of a triangle intersects the other two sides, then it divides those sides proportionally.

Front 72

triangle angle-bisector theorem

Back 72

if a ray bisects an angle angle of a triangle, then it divides the opposite side into segments proportional to the other two sides.